Dushnik-Miller dimension of TD-Delaunay complexes

TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function [5]. In [2] the authors noticed that every triangulation is the TD-Delaunay graph of a set of points in R, and conversely every TD-Delaunay graph is planar. It seems natural to study the generalization of this property in higher dimensions. Such a generalization is obtained by defining an analogue of the triangular distance for R. It is easy to see that TD-Delaunay complexes of Rd−1 are of Dushnik-Miller dimension d. The converse holds for d = 2 or 3 and it was conjectured to hold for larger d [17] (See also [11]). Here we disprove the conjecture already for d = 4.